Exercise 10, § 1

Let $(E_i)_{i \in I}$ be an infinite family of Hausdorff topological vector spaces, none just the point 0, over a non-discrete topological field K. Let E be the direct sum vector space of the $E_i$, and $\mathscr{T}_0$ the topology on E defined in I, p. 24, exerc. 14. Then a subset B of E is bounded for $\mathscr{T}_0$ (III, p. 37, exerc. 1) if and only if B is contained in a product subspace $\prod_{i \in H} E_i$, where H is a finite subset of I and the projections of B on each $E_i$ for $i \in H$ are bounded. Deduce (for $K = \mathbf{R}$ or $\mathbf{C}$) that, if each $E_i$ is a quasi-complete space, then E with $\mathscr{T}_0$ is quasi-complete.